A technician will usually start with the flame or burner, the ignition system, the blower, and the venting, then check the temperature rise across the furnace. On a mobile home unit, expect them to look at the closet's air supply and the duct connections under the floor, which can loosen over time.
If replacement comes up, ask how the new unit will be permitted. Carver lists furnaces among jobs needing an electrical permit, gas or propane hookups need a gas fitting permit, and the Building Department takes these on paper rather than online. The license holder calls in the plumbing, gas, or wiring inspection, and those inspectors work Monday, Wednesday, and Friday.
For homes in a mobile home park, check whether the park association must approve any outdoor work, such as a new vent or a propane tank change, before the job is scheduled.